Frequency Modulation (FM) and Phase Modulation (PM) are two types of analog modulation that are used in various telecommunication and other settings. In FM schemes, an information signal is represented by varying the frequency of a carrier signal. The resulting modulated signal can later be de-modulated (e.g., after transmission) to reconstruct the information signal. According to PM schemes, the information signal is represented by varying the phase of the carrier signal. Because both FM and PM modify angular characteristics of the carrier signal, they generate modulated signals having constant signal envelopes. This makes FM and PM signals less susceptible to noise than other analog modulation techniques, such as Amplitude Modulation (AM). Nonetheless, FM and PM modulated signals are still susceptible to noise including, for example, noise due to channel fading, nonlinear power amplifier characteristics, receiver noise, co-channel interference, adjacent channel interference, etc.
One particular difficulty encountered with FM and PM modulated signals is phase wrapping. For example, FM and PM modulated signals are often demodulated by applying the four quadrant arctangent (tan2−1) to the complex representation of the modulated signal, providing the phase. Because the result of the four quadrant arctangent is limited to the range (π, −π), however, phase signals found using the four quadrant arctangent exhibit discontinuities in applications where the phase varies continuously over a range exceeding (π, −π). The discontinuities are equal to integer multiples of 2π and occur at those time instances when the phase is an odd multiple of π. In order to generate the correct phase, the discontinuities must be removed. In FM schemes, the frequency of the modulated signal is often found by differentiating the phase result of the four quadrant arctangent to provide frequency. After differentiation, the discontinuities in phase manifest as impulse signals in frequency, which are then removed in subsequent processing. In the presence of noise, it can be very difficult to adequately compensate for discontinuities in phase and/or impulses in frequency.